The Fourth-order Bessel–type Differential Equation

The Bessel-type functions, structured as extensions of the classical Bessel functions, were defined by Everitt and Markett in 1994. These special functions are derived by linear combinations and limit processes from the classical orthogonal polynomials, classical Bessel functions and the Krall Jacobi-type and Laguerre-type orthogonal polynomials. These Bessel-type functions are solutions of higher-order linear differential equations, with a regular singularity at the origin and an irregular singularity at the point of infinity of the complex plane.

There is a Bessel-type differential equation for each even-order integer; the equation of order two is the classical Bessel differential equation. These even-order Bessel-type equations are not formal powers of the classical Bessel equation.

When the independent variable of these equations is restricted to the positive real axis of the plane they can be written in the Lagrange symmetric (formally self-adjoint) form of the Glazman–Naimark type, with real coefficients. Embedded in this form of the equation is a spectral parameter; this combination leads to the generation of self-adjoint operators in a weighted Hilbert function space. In the second-order case one of these associated operators has an eigenfunction expansion that leads to the Hankel integral transform.

This article is devoted to a study of the spectral theory of the Bessel-type differential equation of order four; considered on the positive real axis this equation has singularities at both end-points. In the associated Hilbert function space these singular end-points are classified, the minimal and maximal operators are defined and all associated self-adjoint operators are determined, including the Friedrichs self-adjoint operator. The spectral properties of these self-adjoint operators are given in explicit form.

From the properties of the domain of the maximal operator, in the associated Hilbert function space, it is possible to obtain a virial theorem for the fourth-order Bessel-type differential equation.

There are two solutions of this fourth-order equation that can be expressed in terms of classical Bessel functions of order zero and order one. However it appears that additional, independent solutions essentially involve new special functions not yet defined. The spectral properties of the self-adjoint operators suggest that there is an eigenfunction expansion similar to the Hankel transform, but details await a further study of the solutions of the differential equation.     

Singular point perturbation of an odd operator in ℤ2-Graded space

In the paper we study supersymmetric models for point interaction perturbations of operators of Dirac type and their spectral properties. Such models are considered in the class of odd self-adjoint operators in ℤ₂-graded Pontryagin space. We present in detail the previously considered realization method of strongly singular perturbation by means of their embedding into the theory of self-adjoint extensions. We describe odd self-adjoint extensions of odd symmetric operators with deficiency indices (1,1) in ℤ₂-graded Pontryagin space and squares of such extensions using Krein’s formula for the resolvent. The results obtained are refined in application to singular perturbations of odd self-adjoint differential operators. Translated fromMatematicheskie Zametki, Vol. 66, No. 6, pp. 924–940, December, 1999.     

Dissipative operators in the Krein space. Invariant subspaces and properties of restrictions

We prove that a dissipative operator in the Krein space has a maximal nonnegative invariant subspace provided that the operator admits matrix representation with respect to the canonical decomposition of the space and the upper right operator in this representation is compact relative to the lower right operator. Under the additional assumption that the upper and lower left operators are bounded (the so-called Langer condition), this result was proved (in increasing order of generality) by Pontryagin, Krein, Langer, and Azizov. We relax the Langer condition essentially and prove under the new assumptions that a maximal dissipative operator in the Krein space has a maximal nonnegative invariant subspace such that the spectrum of its restriction to this subspace lies in the left half-plane. Sufficient conditions are found for this restriction to be the generator of a holomorphic semigroup or a C ₀-semigroup.     

Second Order Linear Differential Polynomials and Real Meromorphic Functions

The main result determines all real meromorphic functions f of finite lower order in the plane such that f has finitely many zeros and non-real poles, while f′′ + a ₁ f′ + a ₀ f has finitely many non-real zeros, where a ₁ and a ₀ are real rational functions which satisfy a ₁(∞) = 0 and a ₀(x) ≥ 0 for all real x with |x| sufficiently large. This is accomplished by refining some earlier results on the zeros in a neighbourhood of infinity of meromorphic functions and second order linear differential polynomials. Examples are provided illustrating the results.     

Self-adjoint commuting ordinary differential operators

In this paper we study self-adjoint commuting ordinary differential operators of rank two. We find sufficient conditions when an operator of fourth order commuting with an operator of order 4g+2 is self-adjoint. We introduce an equation on potentials V(x),W(x) of the self-adjoint operator \(L=(\partial_{x}^{2}+V)^{2}+W\) and some additional data. With the help of this equation we find the first example of commuting differential operators of rank two corresponding to a spectral curve of higher genus. These operators have polynomial coefficients and define commutative subalgebras of the first Weyl algebra.     

Properties of the solutions of the fourth-order Bessel-type differential equation

The structured Bessel-type functions of arbitrary even-order were introduced by Everitt and Markett in 1994; these functions satisfy linear ordinary differential equations of the same even-order. The differential equations have analytic coefficients and are defined on the whole complex plane with a regular singularity at the origin and an irregular singularity at the point of infinity. They are all natural extensions of the classical second-order Bessel differential equation. Further these differential equations have real-valued coefficients on the positive real half-line of the plane, and can be written in Lagrange symmetric (formally self-adjoint) form. In the fourth-order case, the Lagrange symmetric differential expression generates self-adjoint unbounded operators in certain Hilbert function spaces. These results are recorded in many of the papers here given as references. It is shown in the original paper of 1994 that in this fourth-order case one solution exists which can be represented in terms of the classical Bessel functions of order 0 and 1. The existence of this solution, further aided by computer programs in Maple, led to the existence of a linearly independent basis of solutions of the differential equation. In this paper a new proof of the existence of this solution base is given, on using the advanced theory of special functions in the complex plane. The methods lead to the development of analytical properties of these solutions, in particular the series expansions of all solutions at the regular singularity at the origin of the complex plane.     

A Resolvent Approach to the Real Quantum Plane

Let q ≠ ± 1 be a complex number of modulus one. This paper deals with the operator relation AB = qBA for self-adjoint operators A and B on a Hilbert space. Two classes of well-behaved representations of this relation are studied in detail and characterized by resolvent equations.     

Approximation of isolated eigenvalues of general singular ordinary differential operators

Let A be a self-adjoint operator defined by a general singular ordinary differential expression τ on an interval (a, b), ? ∞ ≤ a < b ≤ ∞. We show that isolated eigenvalues in any gap of the essential spectrum of A are exactly the limits of eigenvalues of suitably chosen self-adjoint realizations A_n of τ on subintervals (a_n, b_n) of (a, b) with a_n → a, b_n → b. This means that eigenvalues of singular ordinary differential operators can be approximated by eigenvalues of regular operators. In the course of the proof we extend a result, which is well known for quasiregular differential expressions, to the general case: If the spectrum of A is not the whole real line, then the boundary conditions needed to define A can be given using solutions of (τ ? λ)u = 0, where λ is contained in the regularity domain of the minimal operator corresponding to τ.     

A unified approach to singular problems arising in the membrane theory

We consider the singular boundary value problem \(({t^n}u't))' + {t^n}f(t,u(t)) = 0,{\rm{ }}\mathop {\lim }\limits_{t \to 0 + } {t^n}u'(t) = 0,{\rm{ }}{a_0}u(1) + {a_1}u'(1 - ) = A,\) where f(t, x) is a given continuous function defined on the set (0, 1]×(0,∞) which can have a time singularity at t = 0 and a space singularity at x = 0. Moreover, n ∈ ?, n ? >2, and a ₀, a ₁, A are real constants such that a ₀ ∈ (0,1), whereas a ₁,A ∈ [0,∞). The main aim of this paper is to discuss the existence of solutions to the above problem and apply the general results to cover certain classes of singular problems arising in the theory of shallow membrane caps, where we are especially interested in characterizing positive solutions. We illustrate the analytical findings by numerical simulations based on polynomial collocation.     

Analysis of a singular functional differential equation that arises from stochastic modeling of population growth

The singular functional differential equation x(1 ? x)A(x)y′(x) + by(h(x)) ? by(x) = ?bg(x), x in (0, 1), is studied for initial data y = 0 on x ? a, y continuous on (a, 1) and y(1?) bounded. The singularity at x = 0+ is removable for a certain class of delayed arguments, h(x). The final end point at x = 1? is the most important singularity because it results in a genuine singular boundary value problem. A formal solution is constructed and is shown to be unique and bounded when g(x) is bounded. A singular decomposition transforms the problem into two nonsingular initial value problems. Singular FDEs of this type arise in the study of the persistence of populations undergoing large random fluctuations when modeled by compound Poisson processes superimposed on logistic-type growth.     

New properties of King’s operators

We prove the existence of a sequence of King’s operators which approximate each continuous function on [0, 1] and preserve the functions e ₀(x) = 1 and e _2i (x) = x ²ⁱ . Moreover, we construct a sequence of polynomial bounded positive linear operators possessing similar properties.     

Left-definite theory with applications to orthogonal polynomials

In the past several years, there has been considerable progress made on a general left-definite theory associated with a self-adjoint operator A that is bounded below in a Hilbert space H; the term ‘left-definite’ has its origins in differential equations but Littlejohn and Wellman [L. L. Littlejohn, R. Wellman, A general left-definite theory for certain self-adjoint operators with applications to differential equations, J. Differential Equations, 181 (2) (2002) 280-339] generalized the main ideas to a general abstract setting. In particular, it is known that such an operator A generates a continuum {H_r}_r>0 of Hilbert spaces and a continuum of {A_r}_r>0 of self-adjoint operators. In this paper, we review the main theoretical results in [L. L. Littlejohn, R. Wellman, A general left-definite theory for certain self-adjoint operators with applications to differential equations, J. Differential Equations, 181 (2) (2002) 280-339]; moreover, we apply these results to several specific examples, including the classical orthogonal polynomials of Laguerre, Hermite, and Jacobi.     

A nonlinear parabolic-Sobolev equation

Let M and L be (nonlinear) operators in a reflexive Banach space B for which Rg(M + L) = B and $\text{¦(Mx ? My) + α(Lx ? Ly)¦ ? | mx ? My |}$ for all α > 0 and pairs x, y in D(M) ∩ D(L). Then there is a unique solution of the Cauchy problem (Mu(t))′ + Lu(t) = 0, Mu(0) = v₀. When M and L are realizations of elliptic partial differential operators in space variables, this gives existence and uniqueness of generalized solutions of boundary value problems for nonlinear partial differential equations of mixed parabolic-Sobolev type.     

Approximating a class of combinatorial problems with rational objective function

In the late seventies, Megiddo proposed a way to use an algorithm for the problem of minimizing a linear function a ₀ + a ₁ x ₁ + . . . + a _n x _n subject to certain constraints to solve the problem of minimizing a rational function of the form (a ₀ + a ₁ x ₁ + . . . + a _n x _n )/(b ₀ + b ₁ x ₁ + . . . + b _n x _n ) subject to the same set of constraints, assuming that the denominator is always positive. Using a rather strong assumption, Hashizume et al. extended Megiddo’s result to include approximation algorithms. Their assumption essentially asks for the existence of good approximation algorithms for optimization problems with possibly negative coefficients in the (linear) objective function, which is rather unusual for most combinatorial problems. In this paper, we present an alternative extension of Megiddo’s result for approximations that avoids this issue and applies to a large class of optimization problems. Specifically, we show that, if there is an α-approximation for the problem of minimizing a nonnegative linear function subject to constraints satisfying a certain increasing property then there is an α-approximation (1/α-approximation) for the problem of minimizing (maximizing) a nonnegative rational function subject to the same constraints. Our framework applies to covering problems and network design problems, among others.     

Fourier-Integral-Operator Approximation of Solutions to First-Order Hyperbolic Pseudodifferential Equations I: Convergence in Sobolev Spaces

An approximation Ansatz for the operator solution, U(z′,z), of a hyperbolic first-order pseudodifferential equation, ? _z  + a(z,x,D _x ) with Re (a) ≥ 0, is constructed as the composition of global Fourier integral operators with complex phases. An estimate of the operator norm in L(H ^(s),H ^(s)) of these operators is provided, which yields a convergence result for the Ansatz to U(z′,z) in some Sobolev space as the number of operators in the composition goes to ∞.     

High Frequency Resolvent Estimates for Perturbations by Large Long-range Magnetic Potentials and Applications to Dispersive Estimates

We prove optimal high-frequency resolvent estimates for self-adjoint operators of the form ${G=\left(i\nabla+b(x)\right)^2+V(x)}$ on ${L^2({\bf R}^n), n\ge 3}$ , where the magnetic potential b(x) and the electric potential V(x) are long-range and large. As an application, we prove dispersive estimates for the wave group ${{\rm e}^{it\sqrt{G}}}$ in the case n = 3 for potentials b(x), V(x) = O(|x|^?2-δ ) for ${|x|\gg 1}$ , where δ > 0.     

Essential self-adjointness of Schrödinger-type operators

The essential self-adjointness of the strongly elliptic operator L = ∑_j,k=1ⁿ (?_j ? ib_j(x)) a_jk(x)(?_k ? ib_k(x)) + q(x) acting on C₀^∞(Rⁿ) is considered, where the matrix (a_jk) is real and symmetric, b_j and q are real, a_jk and b_j are sufficiently smooth, and q?L_loc². It has been shown by Ural'ceva and also Laptev that if q is bounded below and n ? 3 the minimal operator may not be self-adjoint if the principal coefficients rise too rapidly. Thus a theorem of Weyl for ordinary differential operators does not extend to partial differential operators. In this paper it is shown that if q is bounded below and if the principal coefficients are “well behaved” within a sequence of closed shells which go to infinity, then the minimal operator is self-adjoint. It is also shown that a number of results which were known to be true when q is sufficiently smooth may be extended to include the case where q?L_loc². The principal tools used are a distribution inequality due to Tosio Kato and a general maximum principle due to Walter Littman.     

On Olaleru’s open problem on Gregus fixed point theorem

Let (X, d) be a complete metric space and ${TX \longrightarrow X }$ be a mapping with the property d(Tx, Ty) ≤ ad(x, y) + bd(x, Tx) + cd(y, Ty) + ed(y, Tx) + fd(x, Ty) for all ${x, y \in X}$ , where 0 < a < 1, b, c, e, f ≥ 0, a + b + c + e + f = 1 and b + c > 0. We show that if e + f > 0 then T has a unique fixed point and also if e + f ≥ 0 and X is a closed convex subset of a complete metrizable topological vector space (Y, d), then T has a unique fixed point. These results extend the corresponding results which recently obtained in this field. Finally by using our main results we give an answer to the Olaleru’s open problem.     

On Dirichlet Type Problems of Polynomial Dirac Equations with Boundary Conditions

Let ${{\bf D}_{\bf x} := \sum_{i=1}^n \frac{\partial}{\partial x_i} e_i}$ be the Euclidean Dirac operator in ${\mathbb{R}^n}$ and let P(X) = a _m X ^m + . . . + a ₁ X +  a ₀ be a polynomial with real coefficients. Differential equations of the form P(D _x )u(x) = 0 are called homogeneous polynomial Dirac equations with real coefficients. In this paper we treat Dirichlet type problems of the a slightly less general form P(D _x )u(x) = f(x) (where the roots are exclusively real) with prescribed boundary conditions that avoid blow-ups inside the domain. We set up analytic representation formulas for the solutions in terms of hypercomplex integral operators and give exact formulas for the integral kernels in the particular cases dealing with spherical and concentric annular domains. The Maxwell and the Klein–Gordon equation are included as special subcases in this context.